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stop"},{"t":"list_item","d":9,"p":{},"v":"<strong>无法彻底避免</strong>，只能缓解"}]},{"t":"list_item","d":7,"p":{},"v":"欠拟合","c":[{"t":"list_item","d":9,"p":{},"v":"对训练样本的一般特性尚未学好"},{"t":"list_item","d":9,"p":{},"v":"决策树：拓展分支"},{"t":"list_item","d":9,"p":{},"v":"神经网络：增加训练轮数"}]}]}]},{"t":"heading","d":3,"p":{},"v":"评估方法","c":[{"t":"list_item","d":5,"p":{},"v":"通过实验测试来对学习器的泛化误差进行评估并做出选择"},{"t":"list_item","d":5,"p":{},"v":"将测试集上的“测试误差”作为泛化误差的近似"},{"t":"list_item","d":5,"p":{},"v":"测试集要和训练集中的样本尽量互斥"},{"t":"list_item","d":5,"p":{},"v":"获得测试集","c":[{"t":"list_item","d":7,"p":{},"v":"通常将包含m个样本的数据集<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>D</mi><mo>=</mo><mrow><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><mi mathvariant=\"normal\">.</mi><mi mathvariant=\"normal\">.</mi><mi mathvariant=\"normal\">.</mi><mo separator=\"true\">,</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mi>m</mi></msub><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">D={(x_1,y_1),(x_2,y_2),...,(x_m,y_m)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">.</span><span class=\"mord\">.</span><span class=\"mord\">.</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>拆分成训练集S和测试集T"},{"t":"list_item","d":7,"p":{},"v":"在S上训练出模型后，用T来评估其测试误差，作为泛化误差的估计"},{"t":"list_item","d":7,"p":{},"v":"流出法","c":[{"t":"list_item","d":9,"p":{},"v":"直接将数据集划分为两个互斥集合"},{"t":"list_item","d":9,"p":{},"v":"划分尽可能保持数据分布的一致性（分层采样）"},{"t":"list_item","d":9,"p":{},"v":"一般若干次随机划分、重复实验取平均值"},{"t":"list_item","d":9,"p":{},"v":"测试集不能太大、不能太小"},{"t":"list_item","d":9,"p":{},"v":"训练/测试样本比例通常为2:1~4:1"}]},{"t":"list_item","d":7,"p":{},"v":"交叉验证法","c":[{"t":"list_item","d":9,"p":{},"v":"将数据集分层采样划分为k个大小相似的互斥子集，每次用k-1个子集的并集作为训练集，余下的子集作为测试集，最终返回k个此时结果的均值，k最常用的取值是10."},{"t":"list_item","d":9,"p":{},"v":"不受随机样本划分方式的影响"},{"t":"list_item","d":9,"p":{},"v":"结果往往比较准确"},{"t":"list_item","d":9,"p":{},"v":"当数据集比较大时，计算开销难以忍受"}]},{"t":"list_item","d":7,"p":{},"v":"以上两种方法都只用了一部分样本，这必然会引入一些偏差"},{"t":"list_item","d":7,"p":{},"v":"自助法","c":[{"t":"list_item","d":9,"p":{},"v":"自助采样法为基础，对数据集D有放回采样"},{"t":"list_item","d":9,"p":{},"v":"实际模型与预期模型都使用m个训练样本"},{"t":"list_item","d":9,"p":{},"v":"约有1/3的样本没在训练集中出现"},{"t":"list_item","d":9,"p":{},"v":"从促使数据集中产生多个不同的训练集，对集成学习有很大的好处"},{"t":"list_item","d":9,"p":{},"v":"自助法在数据集较小、难以有效划分训练/测试集的时候很有用"},{"t":"list_item","d":9,"p":{},"v":"由于改变了数据集分布可能引入估计偏差。数据量足够多用流出和交叉。"}]}]}]},{"t":"heading","d":3,"p":{},"v":"性能度量","c":[{"t":"list_item","d":5,"p":{},"v":"衡量模型泛化能力的评价标准，反映了任务需求；"},{"t":"list_item","d":5,"p":{},"v":"使用不同的性能度量往往导致不同的评判结果"},{"t":"list_item","d":5,"p":{},"v":"回归任务常用<strong>均方误差</strong>度量"},{"t":"list_item","d":5,"p":{},"v":"错误率：分错样本占总样本比例"},{"t":"list_item","d":5,"p":{},"v":"精度：分对样本占总样本比例"},{"t":"list_item","d":5,"p":{},"v":"TP:真正例、FN：假反例、FP：假正例、TN：真反例"},{"t":"list_item","d":5,"p":{},"v":"查准率：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><mi>T</mi><mi>P</mi></mrow><mrow><mi>T</mi><mi>P</mi><mo>+</mo><mi>F</mi><mi>P</mi></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">P = \\frac{TP}{TP+FP}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.275662em;vertical-align:-0.403331em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.872331em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span><span class=\"mbin mtight\">+</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">F</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"},{"t":"list_item","d":5,"p":{},"v":"查全率：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo>=</mo><mfrac><mrow><mi>T</mi><mi>P</mi></mrow><mrow><mi>T</mi><mi>P</mi><mo>+</mo><mi>F</mi><mi>N</mi></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">P = \\frac{TP}{TP+FN}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.275662em;vertical-align:-0.403331em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.872331em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span><span class=\"mbin mtight\">+</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">F</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"},{"t":"list_item","d":5,"p":{},"v":"查准-查全率曲线：“P-R曲线”","c":[{"t":"list_item","d":7,"p":{},"v":"P-R曲线平衡点F1度量：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>F</mi><mn>1</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mo>∗</mo><mi>P</mi><mo>∗</mo><mi>R</mi></mrow><mrow><mi>P</mi><mo>+</mo><mi>R</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mo>∗</mo><mi>T</mi><mi>P</mi></mrow><mrow><mtext>样例总数</mtext><mo>+</mo><mi>T</mi><mi>P</mi><mo>−</mo><mi>T</mi><mi>N</mi></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">F1 = \\frac{2*P*R}{P+R} = \\frac{2*TP}{样例总数+TP-TN}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">F</span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.275662em;vertical-align:-0.403331em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.872331em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span><span class=\"mbin mtight\">+</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.00773em;\">R</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span><span class=\"mbin mtight\">∗</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span><span class=\"mbin mtight\">∗</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.00773em;\">R</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.275662em;vertical-align:-0.403331em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.872331em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord cjk_fallback mtight\">样</span><span class=\"mord cjk_fallback mtight\">例</span><span class=\"mord cjk_fallback mtight\">总</span><span class=\"mord cjk_fallback mtight\">数</span><span class=\"mbin mtight\">+</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span><span class=\"mbin mtight\">−</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.10903em;\">N</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span><span class=\"mbin mtight\">∗</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"一般形式<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>F</mi><mi>β</mi></msub><mo>=</mo><mfrac><mrow><mo stretchy=\"false\">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy=\"false\">)</mo><mo>∗</mo><mi>P</mi><mo>∗</mo><mi>R</mi></mrow><mrow><mo stretchy=\"false\">(</mo><msup><mi>β</mi><mn>2</mn></msup><mo>∗</mo><mi>P</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mi>R</mi></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">F_{\\beta} = \\frac{(1+\\beta ^2) * P * R}{(\\beta ^2 * P) + R}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.969438em;vertical-align:-0.286108em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">F</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361079999999999em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05278em;\">β</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.62892em;vertical-align:-0.52em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.10892em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05278em;\">β</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7463142857142857em;\"><span style=\"top:-2.786em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mbin mtight\">∗</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span><span class=\"mclose mtight\">)</span><span class=\"mbin mtight\">+</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.00773em;\">R</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">1</span><span class=\"mbin mtight\">+</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05278em;\">β</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8913142857142857em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose mtight\">)</span><span class=\"mbin mtight\">∗</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">P</span><span class=\"mbin mtight\">∗</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.00773em;\">R</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.52em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>","c":[{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>β</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">\\beta = 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05278em;\">β</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span>:标准F1"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>β</mi><mo>&gt;</mo><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">\\beta &gt; 2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05278em;\">β</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">2</span></span></span></span>: 偏重查全率"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>β</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">\\beta &lt; 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05278em;\">β</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&lt;</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span>: 偏重查准率"}]}]},{"t":"list_item","d":5,"p":{},"v":"代价敏感错误率"},{"t":"list_item","d":5,"p":{},"v":"测试性能并不等于泛化误差"},{"t":"list_item","d":5,"p":{},"v":"测试性能随着测试集的变化而变化"},{"t":"list_item","d":5,"p":{},"v":"很多机器学习算法本身有一定的随机性"},{"t":"list_item","d":5,"p":{},"v":"<strong>直接寻去相应评估方法在相应度量下比大小的方法不可取</strong>"}]},{"t":"heading","d":3,"p":{},"v":"比较检验","c":[{"t":"list_item","d":5,"p":{},"v":"t检验"},{"t":"list_item","d":5,"p":{},"v":"Friedman检验"}]},{"t":"heading","d":3,"p":{},"v":"偏差与方差","c":[{"t":"list_item","d":5,"p":{},"v":"帮助解释泛化性能，对学习算法期望的泛化错误率进行拆解"},{"t":"list_item","d":5,"p":{},"v":"对测试样本x，令y_D为x在数据集中的标记，y为x的真实标记f(x;D)为训练集D上学得模型f在x上的预测输出。","c":[{"t":"list_item","d":7,"p":{},"v":"以回归任务为例："},{"t":"list_item","d":7,"p":{},"v":"学习算法的期望预期为 <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">\\overline{f} = \\mathbb{E}_D [f(x;D)]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.08888em;vertical-align:-0.19444em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"使用样本数目相同的不同训练集产生的方差为:<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>v</mi><mi>a</mi><mi>r</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">var(x)=\\mathbb{E}_D[(f(x;D)-\\overline{f}(x))^2]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span><span class=\"mord mathnormal\">a</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"噪声（训练样本的标记与真实标记）为：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>ε</mi><mn>2</mn></msup><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>D</mi></msub><mo>−</mo><mi>y</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">\\varepsilon ^2 = \\mathbb{E}_D[(y_D-y)^2]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">ε</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"期望输出与真实标记的差别称为偏差，即<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>b</mi><mi>i</mi><mi>a</mi><msup><mi>s</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>y</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">bias^2(x)=(\\overline{f}(x)-y)^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">a</span><span class=\"mord\"><span class=\"mord mathnormal\">s</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"假定噪声期望为0:","c":[{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>E</mi><mo stretchy=\"false\">(</mo><mi>f</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><msub><mi>y</mi><mi>D</mi></msub><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">E(f;D) = \\mathbb{E}_D[(f(x;D)-y_D)^2]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mo stretchy=\"false\">(</mo><mo stretchy=\"true\">‾</mo></mover><mi>f</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><msub><mi>y</mi><mi>D</mi></msub><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">= \\mathbb{E}_D[(f(x;D) - \\overline(f)(x) + \\overline{f}(x) - y_D)^2]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2000000000000002em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9500000000000001em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mopen\">(</span></span><span style=\"top:-3.87em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.25em;\"><span></span></span></span></span></span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mclose\">)</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><msub><mi>y</mi><mi>D</mi></msub><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mn>2</mn><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><msub><mi>y</mi><mi>D</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">= \\mathbb{E}_D[(f(x;D) - \\overline{f}(x))^2] + \\mathbb{E}_D[(\\overline{f}(x) - y_D)^2] + \\mathbb{E}_D[2(f(x;D) - \\overline{f}(x))(\\overline{f}(x)-y_D)]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mord\">2</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mclose\">)</span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><msub><mi>y</mi><mi>D</mi></msub><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">= \\mathbb{E}_D[(f(x;D) - \\overline{f}(x))^2] + \\mathbb{E}_D[(\\overline{f}(x) - y_D)^2]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo>:</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>y</mi><mo>+</mo><mi>y</mi><mo>−</mo><msubsup><mi>y</mi><mi>D</mi><mn>2</mn></msubsup><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">= \\mathbb{E}_D[(f(x:D)-\\overline{f}(x))^2] + \\mathbb{E}_D[(\\overline{f}(x)-y+y-y_D^2]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">:</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7777700000000001em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7777700000000001em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.089439em;vertical-align:-0.275331em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-2.424669em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.275331em;\"><span></span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>y</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>y</mi><mo>−</mo><msub><mi>y</mi><mi>D</mi></msub><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><mn>2</mn><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>y</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><mi>y</mi><mo>−</mo><msub><mi>y</mi><mi>D</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">= \\mathbb{E}_D[(f(x;D) - \\overline{f}(x))^2] + \\mathbb{E}_D[(\\overline{f}(x) -y)^2] + \\mathbb{E}_D[(y-y_D)^2] + 2\\mathbb{E}_D[(\\overline{f}(x)-y)(y - y_D)]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\">)</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">]</span></span></span></span>"}]},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>E</mi><mo stretchy=\"false\">(</mo><mi>f</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo><mo>+</mo><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mi>f</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mi>y</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo>+</mo><msub><mi mathvariant=\"double-struck\">E</mi><mi>D</mi></msub><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>D</mi></msub><mo>−</mo><mi>y</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo stretchy=\"false\">]</mo></mrow><annotation encoding=\"application/x-tex\">E(f;D) = \\mathbb{E}_D[(f(x;D) - \\overline{f}(x))^2] + (\\overline{f}(x)-y)^2 + \\mathbb{E}_D[(y_D-y)^2]</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.14444em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.89444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span></span></span><span style=\"top:-3.81444em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathbb\">E</span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">[</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mclose\">]</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>E</mi><mo stretchy=\"false\">(</mo><mi>f</mi><mo separator=\"true\">;</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>b</mi><mi>i</mi><mi>a</mi><msup><mi>s</mi><mn>2</mn></msup><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><mi>v</mi><mi>a</mi><mi>r</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>+</mo><msup><mi>ε</mi><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">E(f;D) = bias^2(x) + var(x) + \\varepsilon ^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">a</span><span class=\"mord\"><span class=\"mord mathnormal\">s</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span><span class=\"mord mathnormal\">a</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8141079999999999em;vertical-align:0em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">ε</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"泛化误差可分解为偏差、方差与噪声之和"},{"t":"list_item","d":7,"p":{},"v":"偏差度量了学习算法期望预测与真实结果的偏离程度；克画了学习算法本身的拟合能力"},{"t":"list_item","d":7,"p":{},"v":"数据扰动所造成的影响"},{"t":"list_item","d":7,"p":{},"v":"学习问题本身的难度"},{"t":"list_item","d":7,"p":{},"v":"泛化性能是由<strong>学习算法的能力、数据的充分性、学习任务本身的难度</strong>所共同决定的"},{"t":"list_item","d":7,"p":{},"v":"偏差-方差窘境","c":[{"t":"list_item","d":9,"p":{},"v":"训练不足时，偏差主导繁华错误率"},{"t":"list_item","d":9,"p":{},"v":"训练程度加深，方差逐渐主导泛化错误率"},{"t":"list_item","d":9,"p":{},"v":"训练充足后，方差主导"}]}]}]}]},{"t":"heading","d":2,"p":{},"v":"线性模型","c":[{"t":"list_item","d":4,"p":{},"v":"基本形式","c":[{"t":"list_item","d":6,"p":{},"v":"线性模型<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mi>w</mi><mn>1</mn></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>w</mi><mi>d</mi></msub><msub><mi>x</mi><mi>d</mi></msub><mo>+</mo><mi>b</mi></mrow><annotation encoding=\"application/x-tex\">f(x)=w_1x_1+w_2x_2+ \\dots +w_dx_d+b</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.66666em;vertical-align:-0.08333em;\"></span><span class=\"minner\">⋯</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">b</span></span></span></span>","c":[{"t":"list_item","d":8,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator=\"true\">;</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator=\"true\">;</mo><msub><mi>x</mi><mn>3</mn></msub><mo separator=\"true\">;</mo><mo>…</mo><mtext> </mtext><mo separator=\"true\">;</mo><msub><mi>x</mi><mi>d</mi></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">x = (x_1;x_2;x_3;\\dots;x_d)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>是由属性描述的示例"},{"t":"list_item","d":8,"p":{},"v":"其中<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding=\"application/x-tex\">x_i</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.58056em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>是x在第i个属性上的取值"}]},{"t":"list_item","d":6,"p":{},"v":"向量形式：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>w</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding=\"application/x-tex\">f(x) = w^Tx+b</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.924661em;vertical-align:-0.08333em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">b</span></span></span></span>","c":[{"t":"list_item","d":8,"p":{},"v":"其中w=(w_1;w_2;\\dots;w_d)"}]}]},{"t":"list_item","d":4,"p":{},"v":"线性模型优点","c":[{"t":"list_item","d":6,"p":{},"v":"形式简单、易于建模，但却蕴含着一些重要的基本思想"},{"t":"list_item","d":6,"p":{},"v":"可解释性强：w直观表达了各属性在预测中的重要性"},{"t":"list_item","d":6,"p":{},"v":"非线性模型的基础","c":[{"t":"list_item","d":8,"p":{},"v":"引入层级结构或高维映射"}]}]},{"t":"list_item","d":4,"p":{},"v":"线性回归","c":[{"t":"list_item","d":6,"p":{},"v":"给定数据集<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>D</mi><mo>=</mo><mrow><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mi>m</mi></msub><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">D={(x_1,y_1),(x_2,y_2),\\dots ,(x_m,y_m)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>","c":[{"t":"list_item","d":8,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>i</mi></msub><mn>1</mn><mo separator=\"true\">;</mo><msub><mi>x</mi><mi>i</mi></msub><mn>2</mn><mo separator=\"true\">;</mo><mo>…</mo><mtext> </mtext><mo separator=\"true\">;</mo><msub><mi>x</mi><mi>i</mi></msub><mi>d</mi><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding=\"application/x-tex\">x_i = (x_i1;x_i2;\\dots;x_id), y_i\\in R</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.58056em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">1</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">2</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathnormal\">d</span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R</span></span></span></span>"}]},{"t":"list_item","d":6,"p":{},"v":"目的：学得一个线性模型以尽可能准确地预测实值输出标记"},{"t":"list_item","d":6,"p":{},"v":"离散属性处理","c":[{"t":"list_item","d":8,"p":{},"v":"有序关系：连续化为连续值"},{"t":"list_item","d":8,"p":{},"v":"无序关系：有K个属性值，则转换为K维向量"}]},{"t":"list_item","d":6,"p":{},"v":"单一属性线性回归：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><mi>x</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>w</mi><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi></mrow><annotation encoding=\"application/x-tex\">f(x)=wx_i+b</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">b</span></span></span></span>使得<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy=\"false\">)</mo><mo>≃</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation encoding=\"application/x-tex\">f(x_i)\\simeq y_i</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≃</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"参数/模型估计：最小二乘法","c":[{"t":"list_item","d":8,"p":{},"v":"最小均方误差"},{"t":"list_item","d":8,"p":{},"v":"分别对w和b求导"},{"t":"list_item","d":8,"p":{},"v":"得到闭式解"}]}]},{"t":"list_item","d":4,"p":{},"v":"多元线性回归","c":[{"t":"list_item","d":6,"p":{},"v":"给定数据集<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>D</mi><mo>=</mo><mrow><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy=\"false\">)</mo><mo separator=\"true\">,</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>m</mi></msub><mo separator=\"true\">,</mo><msub><mi>y</mi><mi>m</mi></msub><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">D={(x_1,y_1),(x_2,y_2,\\dots ,(x_m,y_m))}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mrow><mi>i</mi><mn>1</mn></mrow></msub><mo separator=\"true\">;</mo><msub><mi>x</mi><mrow><mi>i</mi><mn>2</mn></mrow></msub><mo separator=\"true\">;</mo><mo>…</mo><mtext> </mtext><mo separator=\"true\">;</mo><msub><mi>x</mi><mrow><mi>i</mi><mi>d</mi></mrow></msub><mo stretchy=\"false\">)</mo><msub><mi>y</mi><mi>i</mi></msub><mo>∈</mo><mi>R</mi></mrow><annotation encoding=\"application/x-tex\">x_i=(x_{i1};x_{i2};\\dots ;x_{id}) y_i \\in R</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.58056em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mord mtight\">2</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mord mathnormal mtight\">d</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.00773em;\">R</span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"多元线性回归目标：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy=\"false\">)</mo><mo>=</mo><msup><mi>w</mi><mi>T</mi></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi></mrow><annotation encoding=\"application/x-tex\">f(x_i)=w^Tx_i+b</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.991331em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">b</span></span></span></span>使得<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>f</mi><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy=\"false\">)</mo><mo>≃</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation encoding=\"application/x-tex\">f(x_i)\\simeq y_i</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≃</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"把w和b吸收入向量形式<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>w</mi><mo>^</mo></mover><mo>=</mo><mo stretchy=\"false\">(</mo><mi>w</mi><mo separator=\"true\">;</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">\\hat{w} = (w;b)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.69444em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.16666em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mclose\">)</span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"最小二乘法","c":[{"t":"list_item","d":8,"p":{},"v":"同上"},{"t":"list_item","d":8,"p":{},"v":"满秩或正定矩阵可得到确定线性回归模型"},{"t":"list_item","d":8,"p":{},"v":"若<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>X</mi><mi>T</mi></msup><mi>X</mi></mrow><annotation encoding=\"application/x-tex\">X^TX</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8413309999999999em;vertical-align:0em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">X</span></span></span></span>不是满秩矩阵，存在多个解","c":[{"t":"list_item","d":10,"p":{},"v":"根据归纳偏好选择解"},{"t":"list_item","d":10,"p":{},"v":"引入正则化"}]}]}]},{"t":"list_item","d":4,"p":{},"v":"对数线性回归","c":[{"t":"list_item","d":6,"p":{},"v":"输出标记的对数线为线性模型逼近的目标","c":[{"t":"list_item","d":8,"p":{},"v":"虽然形式上时线性回归，但实质上时在求输入空间到输出空间的非线性函数映射"}]},{"t":"list_item","d":6,"p":{},"v":"一般形式<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy=\"false\">(</mo><msup><mi>w</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">y = g^{-1}(w^Tx+b)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0913309999999998em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mclose\">)</span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"g(.)称为联系函数（单调可微函数）"},{"t":"list_item","d":6,"p":{},"v":"对数线性模型回归是<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>g</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mo stretchy=\"false\">)</mo><mo>˙</mo></mover><mo>=</mo><mi>l</mi><mi>n</mi><mo stretchy=\"false\">(</mo><mover accent=\"true\"><mo stretchy=\"false\">)</mo><mo>˙</mo></mover></mrow><annotation encoding=\"application/x-tex\">g(\\dot) = ln(\\dot )</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.23686em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9868600000000001em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mclose\">)</span></span><span style=\"top:-3.319em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.13889em;\"><span class=\"mord\">˙</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.25em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.23686em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mathnormal\">n</span><span class=\"mopen\">(</span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9868600000000001em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mclose\">)</span></span><span style=\"top:-3.319em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.13889em;\"><span class=\"mord\">˙</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.25em;\"><span></span></span></span></span></span></span></span></span>时广义线性模型的特例"},{"t":"list_item","d":6,"p":{},"v":"预测值与输出标记：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>z</mi><mo>=</mo><msup><mi>w</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>∈</mo><mrow><mn>0</mn><mo separator=\"true\">,</mo><mn>1</mn></mrow></mrow><annotation encoding=\"application/x-tex\">z = w^Tx+b y\\in {0,1}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.04398em;\">z</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.924661em;vertical-align:-0.08333em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord\">0</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">1</span></span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"寻找函数将分类标记y与线性回归模型输出z联系起来"},{"t":"list_item","d":6,"p":{},"v":"最理想的函数——单位阶跃函数","c":[{"t":"list_item","d":8,"p":{},"v":"预测值大于0就判为正例，小于0就判为反例，预测值为临界值零则可任意判别"},{"t":"list_item","d":8,"p":{},"v":"缺点：不连续"}]},{"t":"list_item","d":6,"p":{},"v":"替代函数——对数几率函数","c":[{"t":"list_item","d":8,"p":{},"v":"单调可微、任意阶可导"},{"t":"list_item","d":8,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>z</mi></mrow></msup></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">y = \\frac{1}{1+e^{-z}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2484389999999999em;vertical-align:-0.403331em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mbin mtight\">+</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7026642857142857em;\"><span style=\"top:-2.786em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\">z</span></span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"}]}]},{"t":"list_item","d":4,"p":{},"v":"对数几率回归","c":[{"t":"list_item","d":6,"p":{},"v":"运用对数几率函数<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>z</mi></mrow></msup></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">y=\\frac{1}{1+e^{-z}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.2484389999999999em;vertical-align:-0.403331em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mbin mtight\">+</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7026642857142857em;\"><span style=\"top:-2.786em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.04398em;\">z</span></span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.403331em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>变为<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mo stretchy=\"false\">(</mo><msup><mi>w</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy=\"false\">)</mo></mrow></msup></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">y = \\frac{1}{1+e^{-(w^Tx+b)}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.402529em;vertical-align:-0.5574209999999999em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.50091em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mbin mtight\">+</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">e</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9844142857142858em;\"><span style=\"top:-2.9844142857142852em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.698092857142857em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.97733em;\"><span style=\"top:-2.97733em;margin-right:0.1em;\"><span class=\"pstrut\" style=\"height:2.6833299999999998em;\"></span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span><span class=\"mord mathnormal mtight\">x</span><span class=\"mbin mtight\">+</span><span class=\"mord mathnormal mtight\">b</span><span class=\"mclose mtight\">)</span></span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5574209999999999em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"对数几率：样本作为正例的相对可能性的对数：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>l</mi><mi>n</mi><mfrac><mi>y</mi><mrow><mn>1</mn><mo>−</mo><mi>y</mi></mrow></mfrac><mo>=</mo><msup><mi>w</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding=\"application/x-tex\">ln\\frac{y}{1-y}=w^Tx+b</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.228608em;vertical-align:-0.481108em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mathnormal\">n</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7475em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mbin mtight\">−</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.446108em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.481108em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.924661em;vertical-align:-0.08333em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">b</span></span></span></span>"},{"t":"list_item","d":6,"p":{},"v":"优点","c":[{"t":"list_item","d":8,"p":{},"v":"无需事先假设数据分布"},{"t":"list_item","d":8,"p":{},"v":"可得到“类别”的近似概率预测"},{"t":"list_item","d":8,"p":{},"v":"可直接应用现有数值优化算法求取最优解"}]},{"t":"list_item","d":6,"p":{},"v":"<strong>分类学习算法！</strong>"},{"t":"list_item","d":6,"p":{},"v":"极大似然法"},{"t":"list_item","d":6,"p":{},"v":"高阶可导连续凸函数，梯度下降法/牛顿法"}]},{"t":"list_item","d":4,"p":{},"v":"线性判别分析（二分类）","c":[{"t":"list_item","d":6,"p":{},"v":"LDA","c":[{"t":"list_item","d":8,"p":{},"v":"欲使同类样例的投影点尽可能接近，可以让同类样例投影点的协方差尽可能小"},{"t":"list_item","d":8,"p":{},"v":"欲使异类样例的投影点尽可能远离，可以让类中心之间的距离尽可能大"}]},{"t":"list_item","d":6,"p":{},"v":"LDA推广-多分类任务"}]},{"t":"list_item","d":4,"p":{},"v":"多分类学习","c":[{"t":"list_item","d":6,"p":{},"v":"二分类学习方法推广"},{"t":"list_item","d":6,"p":{},"v":"利用二分类学习器-拆解法：拆分策略","c":[{"t":"list_item","d":8,"p":{},"v":"一对一","c":[{"t":"list_item","d":10,"p":{},"v":"拆分阶段","c":[{"t":"list_item","d":12,"p":{},"v":"N个类别两两配对"},{"t":"list_item","d":12,"p":{},"v":"各个二类任务学习分类器"}]},{"t":"list_item","d":10,"p":{},"v":"测试阶段","c":[{"t":"list_item","d":12,"p":{},"v":"新样本提交给所有分类器预测"},{"t":"list_item","d":12,"p":{},"v":"投票产生最终分类结果"}]}]},{"t":"list_item","d":8,"p":{},"v":"一对其余","c":[{"t":"list_item","d":10,"p":{},"v":"任务拆分","c":[{"t":"list_item","d":12,"p":{},"v":"某一类作为正例，其他反例"},{"t":"list_item","d":12,"p":{},"v":"各个二类任务学习分类器"}]},{"t":"list_item","d":10,"p":{},"v":"测试阶段","c":[{"t":"list_item","d":12,"p":{},"v":"新样本提交给所有分类器预测"},{"t":"list_item","d":12,"p":{},"v":"比较各分类器预测置信度"}]}]},{"t":"list_item","d":8,"p":{},"v":"多对多","c":[{"t":"list_item","d":10,"p":{},"v":"若干作为正类，若干类作为反类"},{"t":"list_item","d":10,"p":{},"v":"输出纠错码"},{"t":"list_item","d":10,"p":{},"v":"类别不平衡","c":[{"t":"list_item","d":12,"p":{},"v":"再缩放","c":[{"t":"list_item","d":14,"p":{},"v":"欠采样"},{"t":"list_item","d":14,"p":{},"v":"过采样"},{"t":"list_item","d":14,"p":{},"v":"阈值移动"}]}]}]}]}]}]},{"t":"heading","d":2,"p":{},"v":"决策树","c":[{"t":"heading","d":3,"p":{},"v":"基本流程：基于树结构进行预测","c":[{"t":"list_item","d":5,"p":{},"v":"决策过程中提出的每个判定问题都是对某个属性的“测试”"},{"t":"list_item","d":5,"p":{},"v":"决策过程的最终结论对应了我们所希望的判定结果"},{"t":"list_item","d":5,"p":{},"v":"每个测试的结果或是导出最终结论，或者导出进一步的判定问题，其考虑范围是在上次决策结果的限定范围之内"},{"t":"list_item","d":5,"p":{},"v":"从根节点到每个叶节点的路径对应了一个判定序列"},{"t":"list_item","d":5,"p":{},"v":"决策树学习的目的是为了产生一颗<strong>泛化能力强</strong>，即<strong>处理未见实例能力强的决策树</strong>"}]},{"t":"heading","d":3,"p":{},"v":"划分选择","c":[{"t":"list_item","d":5,"p":{},"v":"决策树学习关键：<strong>选择最优划分属性</strong>。一般而言，随着划分过程不断进行，我们希望决策树的分支节点所包含的样本<strong>尽可能属于同一类别，即结点的“纯度”越来越高</strong>"},{"t":"list_item","d":5,"p":{},"v":"经典的属性划分方法","c":[{"t":"list_item","d":7,"p":{},"v":"<strong>信息增益</strong>","c":[{"t":"list_item","d":9,"p":{},"v":"<strong>信息熵</strong>是度量样本集合纯度最常用的一种指标。"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>E</mi><mi>n</mi><mi>t</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>−</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant=\"normal\">∣</mi><mi>y</mi><mi mathvariant=\"normal\">∣</mi></mrow></msubsup><mrow><msub><mi>p</mi><mi>k</mi></msub><mi>l</mi><mi>o</mi><msub><mi>g</mi><mn>2</mn></msub><msub><mi>p</mi><mi>k</mi></msub></mrow></mrow><annotation encoding=\"application/x-tex\">Ent(D)=-\\sum _{k=1}^{|y|}{p_klog_2p_k}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">t</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.3461079999999999em;vertical-align:-0.30130799999999996em;\"></span><span class=\"mord\">−</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0448em;\"><span style=\"top:-2.398692em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2197999999999998em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord mtight\">∣</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30130799999999996em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mathnormal\">o</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"Ent(D)的值越小，D的纯度越高"},{"t":"list_item","d":9,"p":{},"v":"若p=0，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>p</mi><mi>l</mi><mi>o</mi><msub><mi>g</mi><mn>2</mn></msub><mi>p</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">plog_2p = 0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\">p</span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mathnormal\">o</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathnormal\">p</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">0</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>0</mn><mo>&lt;</mo><mi>E</mi><mi>n</mi><mi>t</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo>&lt;</mo><mi>l</mi><mi>o</mi><msub><mi>g</mi><mn>2</mn></msub><mi mathvariant=\"normal\">∣</mi><mi>y</mi><mi mathvariant=\"normal\">∣</mi></mrow><annotation encoding=\"application/x-tex\">0 &lt; Ent(D &lt; log_2|y|</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68354em;vertical-align:-0.0391em;\"></span><span class=\"mord\">0</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&lt;</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">t</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&lt;</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mathnormal\">o</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord\">∣</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"用属性a对样本集D进行划分所获得的信息增益：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi><mi>a</mi><mi>i</mi><mi>n</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo separator=\"true\">,</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>E</mi><mi>n</mi><mi>t</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><msubsup><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></msubsup><mrow><mfrac><mrow><mi mathvariant=\"normal\">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant=\"normal\">∣</mi></mrow><mrow><mi mathvariant=\"normal\">∣</mi><mi>D</mi><mi mathvariant=\"normal\">∣</mi></mrow></mfrac><mi>E</mi><mi>n</mi><mi>t</mi><mo stretchy=\"false\">(</mo><msup><mi>D</mi><mi>v</mi></msup><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">Gain(D,a)=Ent(D)-\\sum _{v=1}^{V}{\\frac{|D^v|}{|D|} Ent(D^v)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">G</span><span class=\"mord mathnormal\">a</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">a</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">t</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.53em;vertical-align:-0.52em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.981231em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">V</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.01em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"mord mtight\">∣</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span><span class=\"mord mtight\">∣</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.52em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">t</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"信息增益越大，纯度提升越大"},{"t":"list_item","d":9,"p":{},"v":"存在的问题：信息增益对可取值数目较多的属性有所偏好"}]},{"t":"list_item","d":7,"p":{},"v":"增益率","c":[{"t":"list_item","d":9,"p":{},"v":"定义： <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi><mi>a</mi><mi>i</mi><msub><mi>n</mi><mi>r</mi></msub><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo separator=\"true\">,</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mfrac><mrow><mi>G</mi><mi>a</mi><mi>i</mi><mi>n</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo separator=\"true\">,</mo><mi>a</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi>I</mi><mi>V</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">Gain_ratio(D,a)=\\frac{Gain(D,a)}{IV(a)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">G</span><span class=\"mord mathnormal\">a</span><span class=\"mord mathnormal\">i</span><span class=\"mord\"><span class=\"mord mathnormal\">n</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathnormal\">a</span><span class=\"mord mathnormal\">t</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">o</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">a</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.53em;vertical-align:-0.52em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.01em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07847em;\">I</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">V</span><span class=\"mopen mtight\">(</span><span class=\"mord mathnormal mtight\">a</span><span class=\"mclose mtight\">)</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">G</span><span class=\"mord mathnormal mtight\">a</span><span class=\"mord mathnormal mtight\">i</span><span class=\"mord mathnormal mtight\">n</span><span class=\"mopen mtight\">(</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\">a</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.52em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>","c":[{"t":"list_item","d":11,"p":{},"v":"其中<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>I</mi><mi>V</mi><mo stretchy=\"false\">(</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>−</mo><msubsup><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></msubsup><mrow><mfrac><mrow><mi mathvariant=\"normal\">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant=\"normal\">∣</mi></mrow><mrow><mi mathvariant=\"normal\">∣</mi><mi>D</mi><mi mathvariant=\"normal\">∣</mi></mrow></mfrac><mi>l</mi><mi>o</mi><msub><mi>g</mi><mn>2</mn></msub><mfrac><mrow><mi mathvariant=\"normal\">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant=\"normal\">∣</mi></mrow><mrow><mi mathvariant=\"normal\">∣</mi><mi>D</mi><mi mathvariant=\"normal\">∣</mi></mrow></mfrac></mrow></mrow><annotation encoding=\"application/x-tex\">IV(a)=-\\sum _{v=1}^V{\\frac{|D^v|}{|D|}log_2\\frac{|D^v|}{|D|}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07847em;\">I</span><span class=\"mord mathnormal\" style=\"margin-right:0.22222em;\">V</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">a</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.53em;vertical-align:-0.52em;\"></span><span class=\"mord\">−</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.981231em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">V</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.01em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"mord mtight\">∣</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span><span class=\"mord mtight\">∣</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.52em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mathnormal\">o</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.01em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"mord mtight\">∣</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span><span class=\"mord mtight\">∣</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.52em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span></span>"},{"t":"list_item","d":11,"p":{},"v":"称为属性a的“固有值”，属性a的可能取值数目越多，v越大，IV(a)的值通常就越大"}]},{"t":"list_item","d":9,"p":{},"v":"增益率准则对可取值实木较少的属性有所偏好"}]},{"t":"list_item","d":7,"p":{},"v":"基尼指数","c":[{"t":"list_item","d":9,"p":{},"v":"度量数据集D的纯度：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi><mi>i</mi><mi>n</mi><mi>i</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant=\"normal\">∣</mi><mi>y</mi><mi mathvariant=\"normal\">∣</mi></mrow></msubsup><mrow><msub><mo>∑</mo><mrow><msup><mi>k</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup><mo mathvariant=\"normal\">≠</mo><mi>k</mi></mrow></msub><mrow><msub><mi>p</mi><mi>k</mi></msub><msub><mi>p</mi><msup><mi>k</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup></msub></mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi mathvariant=\"normal\">∣</mi><mi>y</mi><mi mathvariant=\"normal\">∣</mi></mrow></msubsup><msubsup><mi>p</mi><mi>k</mi><mn>2</mn></msubsup></mrow><annotation encoding=\"application/x-tex\">Gini(D=\\sum _{k=1}^{|y|}{\\sum _{k&#x27;\\neq k}{p_k p_{k&#x27;}}} = 1- \\sum _{k=1}^{|y|}{p_k^2}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">G</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">i</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.480618em;vertical-align:-0.43581800000000004em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0448em;\"><span style=\"top:-2.398692em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2197999999999998em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord mtight\">∣</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30130799999999996em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.18639799999999984em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6828285714285715em;\"><span style=\"top:-2.786em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span><span class=\"mrel mtight\"><span class=\"mrel mtight\"><span class=\"mord vbox mtight\"><span class=\"thinbox mtight\"><span class=\"rlap mtight\"><span class=\"strut\" style=\"height:0.8888799999999999em;vertical-align:-0.19444em;\"></span><span class=\"inner\"><span class=\"mrel mtight\"></span></span><span class=\"fix\"></span></span></span></span></span><span class=\"mrel mtight\">=</span></span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.43581800000000004em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6828285714285715em;\"><span style=\"top:-2.786em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.3461079999999999em;vertical-align:-0.30130799999999996em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.0448em;\"><span style=\"top:-2.398692em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2197999999999998em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"mord mtight\">∣</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30130799999999996em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2831079999999999em;\"><span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"Gini(D)越小，数据集D的纯度越高"},{"t":"list_item","d":9,"p":{},"v":"属性a的基尼指数定义为：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi><mi>i</mi><mi>n</mi><msub><mi>i</mi><mi>i</mi></msub><mi>n</mi><mi>d</mi><mi>e</mi><mi>x</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo separator=\"true\">,</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></msubsup><mrow><mfrac><mrow><mi mathvariant=\"normal\">∣</mi><msup><mi>D</mi><mi>v</mi></msup><mi mathvariant=\"normal\">∣</mi></mrow><mrow><mi mathvariant=\"normal\">∣</mi><mi>D</mi><mi mathvariant=\"normal\">∣</mi></mrow></mfrac><mi>G</mi><mi>i</mi><mi>n</mi><mi>i</mi><mo stretchy=\"false\">(</mo><msup><mi>D</mi><mi>v</mi></msup><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">Gini_index(D,a)=\\sum _{v=1}^{V}{\\frac{|D^v|}{|D|} Gini(D^v)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">G</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mord\"><span class=\"mord mathnormal\">i</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">d</span><span class=\"mord mathnormal\">e</span><span class=\"mord mathnormal\">x</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">a</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.53em;vertical-align:-0.52em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.981231em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">V</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.01em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"mord mtight\">∣</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">∣</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span><span class=\"mord mtight\">∣</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.52em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord mathnormal\">G</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">i</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"应该选择那个使划分后基尼指数最小的属性作为最优划分属性，即<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>a</mi><mtext>小</mtext></msub><mo>=</mo><munder><mo><mi>arg</mi><mo>⁡</mo><mi>min</mi><mo>⁡</mo></mo><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow></munder><mrow><mi>G</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi mathvariant=\"normal\">_</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>x</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo separator=\"true\">,</mo><mi>a</mi><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">a_{小}=\\mathop{\\arg\\min}\\limits_{a\\in A}{Gini\\_index(D,a)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.58056em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">a</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32833099999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord cjk_fallback mtight\">小</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.716141em;vertical-align:-0.966141em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.66786em;\"><span style=\"top:-2.161229em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">a</span><span class=\"mrel mtight\">∈</span><span class=\"mord mathnormal mtight\">A</span></span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span><span class=\"mop\"><span class=\"mop\">ar<span style=\"margin-right:0.01389em;\">g</span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">min</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.966141em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">G</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">i</span><span class=\"mord\" style=\"margin-right:0.02778em;\">_</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">d</span><span class=\"mord mathnormal\">e</span><span class=\"mord mathnormal\">x</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">a</span><span class=\"mclose\">)</span></span></span></span></span>"}]}]}]},{"t":"heading","d":3,"p":{},"v":"剪枝","c":[{"t":"list_item","d":5,"p":{},"v":"划分选择对决策树的尺寸有较大影响，但对泛化性能的影响有限"},{"t":"list_item","d":5,"p":{},"v":"剪枝方法和程度对决策树泛化性能的影响更为显著"},{"t":"list_item","d":5,"p":{},"v":"对付过拟合的主要手段"},{"t":"list_item","d":5,"p":{},"v":"基本策略","c":[{"t":"list_item","d":7,"p":{},"v":"预剪枝"},{"t":"list_item","d":7,"p":{},"v":"后剪枝"}]},{"t":"list_item","d":5,"p":{},"v":"留出法：预留一部分数据用作验证机，进行性能评估"},{"t":"list_item","d":5,"p":{},"v":"预剪枝","c":[{"t":"list_item","d":7,"p":{},"v":"优点","c":[{"t":"list_item","d":9,"p":{},"v":"降低过拟合风险"},{"t":"list_item","d":9,"p":{},"v":"显著减少训练/测试时间开销"}]},{"t":"list_item","d":7,"p":{},"v":"缺点","c":[{"t":"list_item","d":9,"p":{},"v":"欠拟合风险"}]}]},{"t":"list_item","d":5,"p":{},"v":"后剪枝","c":[{"t":"list_item","d":7,"p":{},"v":"优点： 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每层神经元与下一层神经元全互联，神经元直接不存在同层链接或跨层链接"},{"t":"list_item","d":9,"p":{},"v":"前馈：输出层接受外界输入，隐含层与输出层神经元对信号进行加工，最终输出层输出"},{"t":"list_item","d":9,"p":{},"v":"学习：根据训练数据来调整神经元之间的“链接权”"},{"t":"list_item","d":9,"p":{},"v":"多层网络：包含隐含层的网络"},{"t":"list_item","d":9,"p":{},"v":"强大的表示能力：只需要一个包含足够多神经元的隐层，就能以任意进度逼近任意复杂度的连续函数。"}]}]},{"t":"list_item","d":5,"p":{},"v":"误差逆传播算法-BP算法","c":[{"t":"list_item","d":7,"p":{},"v":"最成功、最常用的训练多层前馈神经网络的学习算法，可被用于多种任务。"},{"t":"list_item","d":7,"p":{},"v":"前向计算"},{"t":"list_item","d":7,"p":{},"v":"参数数目"},{"t":"list_item","d":7,"p":{},"v":"参数优化"},{"t":"list_item","d":7,"p":{},"v":"bp算法基于梯度下降策略，以目标的负梯度方向对参数进行调整。对误差<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>E</mi><mi>k</mi></msub></mrow><annotation encoding=\"application/x-tex\">E_k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>，给定学习率n"},{"t":"list_item","d":7,"p":{},"v":"链式法则"}]},{"t":"list_item","d":5,"p":{},"v":"全局最小与局部最小","c":[{"t":"list_item","d":7,"p":{},"v":"多组不同初始参数优化神经网络，选取误差最小的解作为最终参数"},{"t":"list_item","d":7,"p":{},"v":"模拟退火技术"},{"t":"list_item","d":7,"p":{},"v":"随机梯度下降"},{"t":"list_item","d":7,"p":{},"v":"遗传算法"},{"t":"list_item","d":7,"p":{},"v":"梯度消散"},{"t":"list_item","d":7,"p":{},"v":"局部极值"},{"t":"list_item","d":7,"p":{},"v":"训练数据与运算能力"},{"t":"list_item","d":7,"p":{},"v":"梯度消失"}]},{"t":"list_item","d":5,"p":{},"v":"RBF网络","c":[{"t":"list_item","d":7,"p":{},"v":"一种单隐层前馈神经网络，它使用径向基函数作为隐层神经元激活函数，而输出层则是隐层神经元输出的线性组合。"},{"t":"list_item","d":7,"p":{},"v":"性质：具有足够多隐层神经元RBF神经网络能以任意精度逼近任意连续函数"},{"t":"list_item","d":7,"p":{},"v":"RBF网络训练","c":[{"t":"list_item","d":9,"p":{},"v":"确定神经元中心，常用的方式包括随机采样、聚类等"},{"t":"list_item","d":9,"p":{},"v":"利用BP算法等确定参数"}]}]},{"t":"list_item","d":5,"p":{},"v":"ART网络","c":[{"t":"list_item","d":7,"p":{},"v":"竞争学习：一种常用的<strong>无监督学习策略</strong>，在使用该策略时，网络的输出神经元互相竞争，每一时刻仅有一个神经元被激活，其他神经元的状态被抑制。"},{"t":"list_item","d":7,"p":{},"v":"是竞争学习的重要代表"},{"t":"list_item","d":7,"p":{},"v":"由比较层、识别层、识别阈值和重置模块构成"},{"t":"list_item","d":7,"p":{},"v":"比较层负责接收输入样本，并将其传送给识别层神经元"},{"t":"list_item","d":7,"p":{},"v":"识别层每个神经元对应一个模式类，神经元数目可在训练过程中动态增长以增加新的模式类"},{"t":"list_item","d":7,"p":{},"v":"网络性能依赖于识别阈值"},{"t":"list_item","d":7,"p":{},"v":"较好的解决了竞争学习中的“可塑性-稳定性窘境”","c":[{"t":"list_item","d":9,"p":{},"v":"可塑性：神经网络要有学习新知识的能力"},{"t":"list_item","d":9,"p":{},"v":"稳定性：神经网络在学习新知识时要保持对旧知识的记忆"}]},{"t":"list_item","d":7,"p":{},"v":"可以增量学习或在线学习"},{"t":"list_item","d":7,"p":{},"v":"发展：ART2\\FuzART\\ARTMAP"}]},{"t":"list_item","d":5,"p":{},"v":"SOM网络"},{"t":"list_item","d":5,"p":{},"v":"Elman网络"},{"t":"list_item","d":5,"p":{},"v":"Boltzmann机"},{"t":"list_item","d":5,"p":{},"v":"RBM机训练"},{"t":"list_item","d":5,"p":{},"v":"自编码器","c":[{"t":"list_item","d":7,"p":{},"v":"一种尽量复现输入信号的神经网络"},{"t":"list_item","d":7,"p":{},"v":"自编码器必须捕捉可以代表输入数据的最重要的特征；"},{"t":"list_item","d":7,"p":{},"v":"无监督学习方法，可以用于数据降维及其特征抽取"}]},{"t":"list_item","d":5,"p":{},"v":"深度自编码器：由两个对称的深度置信网络组成"}]},{"t":"heading","d":3,"p":{},"v":"深度学习模型","c":[{"t":"list_item","d":5,"p":{},"v":"卷积神经网络CNN","c":[{"t":"list_item","d":7,"p":{},"v":"人工神经网络的一种，语音分析、图像识别。其权值共享网络结构使之更类似于生物神经网络"},{"t":"list_item","d":7,"p":{},"v":"降低了网络模型的复杂度，减少了权值的数量"},{"t":"list_item","d":7,"p":{},"v":"优点在网络输入时多维图像时表现得尤为明显，使图像可以直接作为网络得输入，避免了传统识别算法中复杂得特征提取和数据重建过程"},{"t":"list_item","d":7,"p":{},"v":"卷积网络使为死别二维形状而特殊设计的一个多层感知器，这种网络结构对平移、比例缩放、倾斜或者其它形式的变形具有高度不变性。"},{"t":"list_item","d":7,"p":{},"v":"核心思想","c":[{"t":"list_item","d":9,"p":{},"v":"局部感受野：图像的一个关键特征使，相邻的两个像素通常比远距离的两个像素有更强的相关性"},{"t":"list_item","d":9,"p":{},"v":"权值共享：图像某个区域有用的局部特征对其他区域也是有用的，因为目标可以出现在图像中不同的位置"},{"t":"list_item","d":9,"p":{},"v":"降采样：卷积提取的局部特征可以在后续的层中进行聚合，从而使得卷积神经网络具有一定的空间不变性，实现一定程度的平移和变形的不变性。"}]},{"t":"list_item","d":7,"p":{},"v":"Convolve卷积","c":[{"t":"list_item","d":9,"p":{},"v":"卷积使两个变量在某范围内相乘后求和的结果"},{"t":"list_item","d":9,"p":{},"v":"卷积核"},{"t":"list_item","d":9,"p":{},"v":"卷积层"},{"t":"list_item","d":9,"p":{},"v":"参数减少与权值共享"},{"t":"list_item","d":9,"p":{},"v":"卷积网络：每个过滤器获取一个不同的信号"},{"t":"list_item","d":9,"p":{},"v":"池化层：对图像进行池化，可以引入平移、遮掩不变性，减少输入数据对网络的影响，增强网络的鲁棒性","c":[{"t":"list_item","d":11,"p":{},"v":"降采样/池化：可以对图像某个区域上的特征取平均值。这种聚合的操作叫做池化"}]}]},{"t":"list_item","d":7,"p":{},"v":"架构","c":[{"t":"list_item","d":9,"p":{},"v":"input image"},{"t":"list_item","d":9,"p":{},"v":"Convoluion(Learned)"},{"t":"list_item","d":9,"p":{},"v":"Non-linearity"},{"t":"list_item","d":9,"p":{},"v":"Pooling"},{"t":"list_item","d":9,"p":{},"v":"Feature maps"}]}]},{"t":"list_item","d":5,"p":{},"v":"循环神经网络（RNN）","c":[{"t":"list_item","d":7,"p":{},"v":"通过使用带自反馈的神经元，能够处理任意长度的序列。"},{"t":"list_item","d":7,"p":{},"v":"比前馈神经网络更加符合生物神经网络结构。"},{"t":"list_item","d":7,"p":{},"v":"被广泛应用在语音识别、语言模型以及自然语言生成等任务上"},{"t":"list_item","d":7,"p":{},"v":"一个完全连接的循环网络是任何非线性动力系统的近似器"},{"t":"list_item","d":7,"p":{},"v":"作用：","c":[{"t":"list_item","d":9,"p":{},"v":"输入-输出映射（机器学习）"},{"t":"list_item","d":9,"p":{},"v":"存储"}]},{"t":"list_item","d":7,"p":{},"v":"序列到类别：输入到输出"},{"t":"list_item","d":7,"p":{},"v":"同步的序列到序列模式"},{"t":"list_item","d":7,"p":{},"v":"异步的序列到序列模式"},{"t":"list_item","d":7,"p":{},"v":"堆叠循环神经网络"},{"t":"list_item","d":7,"p":{},"v":"双向循环神经网络"},{"t":"list_item","d":7,"p":{},"v":"使用BPTT算法训练"}]},{"t":"list_item","d":5,"p":{},"v":"生成对抗网络GAN","c":[{"t":"list_item","d":7,"p":{},"v":"模型能够用来从学习到的分布产生出新样本"},{"t":"list_item","d":7,"p":{},"v":"解决密度估计，这一非监督学习中的核心问题"},{"t":"list_item","d":7,"p":{},"v":"显示密度模型","c":[{"t":"list_item","d":9,"p":{},"v":"显示的构建出样本的密度函数p(x|\\theta) 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class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>P</mi><mi>g</mi></msub><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">P_g(G)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.036108em;vertical-align:-0.286108em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15139200000000003em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">g</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">G</span><span class=\"mclose\">)</span></span></span></span>没有显示的表达"},{"t":"list_item","d":9,"p":{},"v":"比较难训练，D与G之间需要很好的同步，例如D更新k次而G更新1次"}]}]}]}]},{"t":"heading","d":2,"p":{},"v":"支持向量机","c":[{"t":"heading","d":3,"p":{},"v":"间隔与支持向量","c":[{"t":"list_item","d":5,"p":{},"v":"超平面方程<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>w</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">w^Tx+b=0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.924661em;vertical-align:-0.08333em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">0</span></span></span></span>"},{"t":"list_item","d":5,"p":{},"v":"y =","c":[{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>1</mn><mo separator=\"true\">,</mo><mi>i</mi><mi>f</mi><mover accent=\"true\"><mi>w</mi><mo>⃗</mo></mover><mo>⋅</mo><mover accent=\"true\"><mi>x</mi><mo>⃗</mo></mover><mo>+</mo><mi>b</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">1, if\\vec{w}\\cdot\\vec{x}+b \\ge 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord\">1</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">⋅</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.79733em;vertical-align:-0.08333em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.83041em;vertical-align:-0.13597em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≥</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>−</mo><mn>1</mn><mo separator=\"true\">,</mo><mi>i</mi><mi>f</mi><mover accent=\"true\"><mi>w</mi><mo>⃗</mo></mover><mo>⋅</mo><mover accent=\"true\"><mi>x</mi><mo>⃗</mo></mover><mo>+</mo><mi>b</mi><mo>≤</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">-1, if\\vec{w}\\cdot\\vec{x}+b \\leq -1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9084399999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord\">−</span><span class=\"mord\">1</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\" style=\"margin-right:0.10764em;\">f</span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.15216em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">⋅</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.79733em;vertical-align:-0.08333em;\"></span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.714em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.20772em;\"><span class=\"overlay\" style=\"height:0.714em;width:0.471em;\"><svg width='0.471em' height='0.714em' style='width:0.471em' viewBox='0 0 471 714' preserveAspectRatio='xMinYMin'><path d='M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z'/></svg></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.83041em;vertical-align:-0.13597em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≤</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">1</span></span></span></span>"}]},{"t":"list_item","d":5,"p":{},"v":"支持向量机基本型","c":[{"t":"list_item","d":7,"p":{},"v":"最大化间隔：寻找参数|w|和|b|使得<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi mathvariant=\"normal\">∣</mi><mi>γ</mi><mi mathvariant=\"normal\">∣</mi></mrow><annotation encoding=\"application/x-tex\">|\\gamma|</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.05556em;\">γ</span><span class=\"mord\">∣</span></span></span></span>最大"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><munder><mo><mi>arg</mi><mo>⁡</mo><mi>max</mi><mo>⁡</mo></mo><mrow><mi>w</mi><mo separator=\"true\">,</mo><mi>b</mi></mrow></munder><mfrac><mn>2</mn><mrow><mo stretchy=\"false\">∥</mo><mi>w</mi><mo stretchy=\"false\">∥</mo></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">\\mathop{\\arg\\max}\\limits_{w,b}{\\frac{2}{\\lVert w\\rVert}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.927764em;vertical-align:-1.082656em;\"></span><span class=\"mop op-limits\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.43055999999999983em;\"><span style=\"top:-2.153452em;margin-left:0em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\">b</span></span></span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span><span class=\"mop\"><span class=\"mop\">ar<span style=\"margin-right:0.01389em;\">g</span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\">max</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.082656em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.655em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">∥</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose mtight\">∥</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.52em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>s</mi><mi mathvariant=\"normal\">.</mi><mi>t</mi><mi mathvariant=\"normal\">.</mi><msub><mi>y</mi><mi>i</mi></msub><mo stretchy=\"false\">(</mo><msup><mi>w</mi><mi>T</mi></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo stretchy=\"false\">)</mo><mo>≥</mo><mn>1</mn><mo separator=\"true\">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator=\"true\">,</mo><mn>2</mn><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><mi>m</mi><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">s.t. y_i(w^Tx_i+b) \\ge 1, i = 1,2,\\dots,m.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0913309999999998em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">s</span><span class=\"mord\">.</span><span class=\"mord mathnormal\">t</span><span class=\"mord\">.</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≥</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.85396em;vertical-align:-0.19444em;\"></span><span class=\"mord\">1</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"></span><span class=\"mord\">1</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">2</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">m</span><span class=\"mord\">.</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"$\\mathop{\\arg\\min}\\limits_{w,b}{\\frac{1}{2}\\lVert w\\rVert ^2$"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>s</mi><mi mathvariant=\"normal\">.</mi><mi>t</mi><mi mathvariant=\"normal\">.</mi><msub><mi>y</mi><mi>i</mi></msub><mo stretchy=\"false\">(</mo><msup><mi>w</mi><mi>T</mi></msup><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo stretchy=\"false\">)</mo><mo>≥</mo><mn>1</mn><mo separator=\"true\">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator=\"true\">,</mo><mn>2</mn><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><mi>m</mi><mi mathvariant=\"normal\">.</mi></mrow><annotation encoding=\"application/x-tex\">s.t. y_i(w^Tx_i+b) \\ge 1, i = 1,2,\\dots,m.</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0913309999999998em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">s</span><span class=\"mord\">.</span><span class=\"mord mathnormal\">t</span><span class=\"mord\">.</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≥</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.85396em;vertical-align:-0.19444em;\"></span><span class=\"mord\">1</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">i</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"></span><span class=\"mord\">1</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\">2</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">m</span><span class=\"mord\">.</span></span></span></span>"}]},{"t":"list_item","d":5,"p":{},"v":"支持向量机求解","c":[{"t":"list_item","d":7,"p":{},"v":"凸二次规划"}]}]},{"t":"heading","d":3,"p":{},"v":"对偶问题"},{"t":"heading","d":3,"p":{},"v":"核函数"},{"t":"heading","d":3,"p":{},"v":"软间隔与正则化"},{"t":"heading","d":3,"p":{},"v":"支持向量回归"},{"t":"heading","d":3,"p":{},"v":"核方法"}]},{"t":"heading","d":2,"p":{},"v":"贝叶斯分类器","c":[{"t":"heading","d":3,"p":{},"v":"贝叶斯决策论"},{"t":"heading","d":3,"p":{},"v":"极大似然估计"},{"t":"heading","d":3,"p":{},"v":"朴素贝叶斯分类器","c":[{"t":"list_item","d":5,"p":{},"v":"估计后验概率主要困难：类条件概率P(x|c)是<strong>所有属性上的联合概率难以从有限的训练样本估计获得</strong>。"},{"t":"list_item","d":5,"p":{},"v":"属性条件独立性假设"},{"t":"list_item","d":5,"p":{},"v":"贝叶斯分类实例","c":[{"t":"list_item","d":7,"p":{},"v":"首先估计类先验概率P(c)"},{"t":"list_item","d":7,"p":{},"v":"为每个属性估计条件概率P（x_i|c）"},{"t":"list_item","d":7,"p":{},"v":"分别对样本每个属性为正例的概率求累乘，再为反例的概率求累乘"},{"t":"list_item","d":7,"p":{},"v":"对比2概率积，大的一方为贝叶斯分类器的预测"}]},{"t":"list_item","d":5,"p":{},"v":"避免其他属性携带的信息被训练集中未出现的属性值抹去（被乘了0），用拉普拉斯修正。","c":[{"t":"list_item","d":7,"p":{},"v":"添加了可能取值项，避免了分子为0的情况"}]},{"t":"list_item","d":5,"p":{},"v":"对预测速度要求高：预计计算所有概率估计值，使用时“查表”"},{"t":"list_item","d":5,"p":{},"v":"任务数据更替平凡：不进行任何训练，收到预测请求时再估值（懒惰学习，lazy learning）"},{"t":"list_item","d":5,"p":{},"v":"数据不断增加：基于现有估值，对新样本涉及的概率估值进行修正（增量学习）"}]},{"t":"heading","d":3,"p":{},"v":"半朴素贝叶斯分类器"},{"t":"heading","d":3,"p":{},"v":"贝叶斯网"},{"t":"heading","d":3,"p":{},"v":"EM算法"}]},{"t":"heading","d":2,"p":{},"v":"聚类","c":[{"t":"heading","d":3,"p":{},"v":"无监督学习"},{"t":"heading","d":3,"p":{},"v":"聚类任务","c":[{"t":"list_item","d":5,"p":{},"v":"无监督学习"},{"t":"list_item","d":5,"p":{},"v":"聚类目标：将数据集中的样本划分为若干个通常不相交的子集"},{"t":"list_item","d":5,"p":{},"v":"聚类既可以作为一个单独过程，也可以为分类等其他学习任务的前驱过程"}]},{"t":"heading","d":3,"p":{},"v":"性能度量","c":[{"t":"list_item","d":5,"p":{},"v":"有效性指标：同一簇的样本尽可能彼此相似，不同簇的样本尽可能不同。-》“簇内相似度”高。“簇间相似度”低。"},{"t":"list_item","d":5,"p":{},"v":"外部指标： 将聚类结果与某个参考模型进行比较","c":[{"t":"list_item","d":7,"p":{},"v":"Jaccard系数"},{"t":"list_item","d":7,"p":{},"v":"FM系数"},{"t":"list_item","d":7,"p":{},"v":"Rand指数"},{"t":"list_item","d":7,"p":{},"v":"<strong>【0，1】区间内，越大越好</strong>"}]},{"t":"list_item","d":5,"p":{},"v":"内部指标：直接考察聚类结果而不用任何参考模型","c":[{"t":"list_item","d":7,"p":{},"v":"簇c内样本间的平均距离"},{"t":"list_item","d":7,"p":{},"v":"最远距离"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mtext>簇</mtext><msub><mi>C</mi><mi>i</mi></msub><mtext>与</mtext><msub><mi>C</mi><mi>j</mi></msub><mtext>最近样本间的距离</mtext></mrow><annotation encoding=\"application/x-tex\">簇C_i与C_j最近样本间的距离</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.969438em;vertical-align:-0.286108em;\"></span><span class=\"mord cjk_fallback\">簇</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord cjk_fallback\">与</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.311664em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span></span></span></span></span></span></span><span class=\"mord cjk_fallback\">最</span><span class=\"mord cjk_fallback\">近</span><span class=\"mord cjk_fallback\">样</span><span class=\"mord cjk_fallback\">本</span><span class=\"mord cjk_fallback\">间</span><span class=\"mord cjk_fallback\">的</span><span class=\"mord cjk_fallback\">距</span><span class=\"mord cjk_fallback\">离</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mtext>簇</mtext><msub><mi>C</mi><mi>i</mi></msub><mtext>与簇</mtext><msub><mi>C</mi><mi>j</mi></msub><mtext>中心点间的距离</mtext></mrow><annotation encoding=\"application/x-tex\">簇C_i与簇C_j中心点间的距离</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.969438em;vertical-align:-0.286108em;\"></span><span class=\"mord cjk_fallback\">簇</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord cjk_fallback\">与</span><span class=\"mord cjk_fallback\">簇</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.311664em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span></span></span></span></span></span></span><span class=\"mord cjk_fallback\">中</span><span class=\"mord cjk_fallback\">心</span><span class=\"mord cjk_fallback\">点</span><span class=\"mord cjk_fallback\">间</span><span class=\"mord cjk_fallback\">的</span><span class=\"mord cjk_fallback\">距</span><span class=\"mord cjk_fallback\">离</span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"DB指数：越小越好"},{"t":"list_item","d":7,"p":{},"v":"Dunn指数：越大越好"}]},{"t":"list_item","d":5,"p":{},"v":"<strong>聚类的好坏不存在绝对标准</strong>"}]},{"t":"heading","d":3,"p":{},"v":"距离计算","c":[{"t":"list_item","d":5,"p":{},"v":"聚类度量的性质","c":[{"t":"list_item","d":7,"p":{},"v":"非负性"},{"t":"list_item","d":7,"p":{},"v":"同一性"},{"t":"list_item","d":7,"p":{},"v":"对称性"},{"t":"list_item","d":7,"p":{},"v":"直递性"}]},{"t":"list_item","d":5,"p":{},"v":"常用距离","c":[{"t":"list_item","d":7,"p":{},"v":"闵可夫斯基距离"},{"t":"list_item","d":7,"p":{},"v":"欧式距离"},{"t":"list_item","d":7,"p":{},"v":"曼哈顿距离"}]},{"t":"list_item","d":5,"p":{},"v":"属性","c":[{"t":"list_item","d":7,"p":{},"v":"连续属性：定义域上有无穷多个可能的取值"},{"t":"list_item","d":7,"p":{},"v":"离散属性：定义域上是有限个可能的取值"},{"t":"list_item","d":7,"p":{},"v":"有序属性：闵可夫斯基距离（{1，2，3}1比2近，又比三远）"},{"t":"list_item","d":7,"p":{},"v":"无序属性：VDM（{飞机，火车，轮船}）"}]},{"t":"list_item","d":5,"p":{},"v":"距离度量","c":[{"t":"list_item","d":7,"p":{},"v":"VDM（处理无序属性）"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>M</mi><mi>i</mi><mi>n</mi><mi>k</mi><mi>o</mi><mi>v</mi><mi>D</mi><msub><mi>M</mi><mi>p</mi></msub></mrow><annotation encoding=\"application/x-tex\">MinkovDM_p</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.980548em;vertical-align:-0.286108em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">M</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"mord mathnormal\">o</span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">v</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.10903em;\">M</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15139200000000003em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286108em;\"><span></span></span></span></span></span></span></span></span></span>（处理混合属性）"},{"t":"list_item","d":7,"p":{},"v":"加权距离"}]}]},{"t":"heading","d":3,"p":{},"v":"原型聚类","c":[{"t":"list_item","d":5,"p":{},"v":"基于原型的聚类，假设聚类结构能通过一组原型刻画"},{"t":"list_item","d":5,"p":{},"v":"算法过程","c":[{"t":"list_item","d":7,"p":{},"v":"先对原型进行初始化，再对原型进行迭代更新求解。"},{"t":"list_item","d":7,"p":{},"v":"采用不同的原型表示，参数不同的算法"}]},{"t":"list_item","d":5,"p":{},"v":"几种著名算法","c":[{"t":"list_item","d":7,"p":{},"v":"k-means算法","c":[{"t":"list_item","d":9,"p":{},"v":"给定数据集<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>D</mi><mo>=</mo><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><msub><mi>x</mi><mi>m</mi></msub></mrow></mrow><annotation encoding=\"application/x-tex\">D={x_1,x_2,\\dots ,x_m}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>,k均值算法针对聚类所得簇划分<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>C</mi><mo>=</mo><mrow><msub><mi>C</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>C</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><msub><mi>C</mi><mi>k</mi></msub></mrow></mrow><annotation encoding=\"application/x-tex\">C={C_1,C_2,\\dots ,C_k}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8777699999999999em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>最小化平方误差"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>E</mi><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></msubsup><mrow><msub><mo>∑</mo><mrow><mi>x</mi><mo>∈</mo><msub><mi>C</mi><mi>j</mi></msub></mrow></msub><mrow><mo stretchy=\"false\">∥</mo><mi>x</mi><mo>−</mo><msub><mi>μ</mi><mi>i</mi></msub><msubsup><mo stretchy=\"false\">∥</mo><mn>2</mn><mn>2</mn></msubsup></mrow></mrow></mrow><annotation encoding=\"application/x-tex\">E=\\sum _{i=1}^k{\\sum _{x\\in C_j}{\\lVert x-\\mu _i\\rVert _2^2}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.68333em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.486038em;vertical-align:-0.49703em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9890079999999999em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.17862099999999997em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"mrel mtight\">∈</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:-0.07153em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.05724em;\">j</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2818857142857143em;\"><span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.49703em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mopen\">∥</span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">∥</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.24810799999999997em;\"><span></span></span></span></span></span></span></span></span></span></span></span>","c":[{"t":"list_item","d":11,"p":{},"v":"其中，<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>μ</mi><mi>i</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\mu _i</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">μ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>是簇<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding=\"application/x-tex\">C_i</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.83333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.07153em;\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>得均值向量。"},{"t":"list_item","d":11,"p":{},"v":"E值再一定程度上刻画了簇内样本围绕簇均值向量得紧密程度，E值越小，则簇内样本相似度越高"}]},{"t":"list_item","d":9,"p":{},"v":"算法流程","c":[{"t":"bullet_list","d":10,"p":{},"v":"","c":[{"t":"list_item","d":11,"p":{},"v":"repeat"}]},{"t":"ordered_list","d":10,"p":{"start":1},"v":"","c":[{"t":"list_item","d":11,"p":{"index":1},"v":"1. （更新）簇划分；"},{"t":"list_item","d":11,"p":{"index":2},"v":"2. 计算每个簇得均值向量"}]},{"t":"bullet_list","d":10,"p":{},"v":"","c":[{"t":"list_item","d":11,"p":{},"v":"until 当前均值向量均未更新"}]}]},{"t":"list_item","d":9,"p":{},"v":"每次迭代都会得到当前簇最短平均距离的中心点，如果集合内某点到其簇内中心比其他簇中心要长，那么我们认为该点不属于该簇。"}]},{"t":"list_item","d":7,"p":{},"v":"学习向量量化算法"},{"t":"list_item","d":7,"p":{},"v":"高斯混合聚类算法"}]}]}]},{"t":"heading","d":2,"p":{},"v":"降维与度量学习","c":[{"t":"heading","d":3,"p":{},"v":"k近邻学习","c":[{"t":"list_item","d":5,"p":{},"v":"一种常用的监督学习方法","c":[{"t":"list_item","d":7,"p":{},"v":"确定训练样本，以及某种距离度量"},{"t":"list_item","d":7,"p":{},"v":"对于某给定的测试样本，找到训练集中距离最近的k个样本，对于分类问题使用“投票法”获得预测结果，对于回归问题使用“平均法”获得预测结果。","c":[{"t":"list_item","d":9,"p":{},"v":"投票法：选择k个样本中出现最多的类别标记作为预测结果。"},{"t":"list_item","d":9,"p":{},"v":"平均法：将这k个样本的实值输出标记的平均值作为预测结果。"}]}]},{"t":"list_item","d":5,"p":{},"v":"懒惰学习：此类学习技术在训练阶段仅保存样本，训练时间开销为0，待收到测试样本后再进行处理"},{"t":"list_item","d":5,"p":{},"v":"急切学习：训练阶段就对样本进行学习和处理的方法"},{"t":"list_item","d":5,"p":{},"v":"k近邻分类器中的k为重要参数，k取不同值，分类结果会显著不同。采用不同距离计算方式，找出的近邻可能有显著差别，导致分类结果有显著不同"},{"t":"list_item","d":5,"p":{},"v":"最近邻分类器（1NN，即k=1）","c":[{"t":"list_item","d":7,"p":{},"v":"错误率：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>e</mi><mi>r</mi><mi>r</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><msub><mo>∑</mo><mrow><mi>c</mi><mo>∈</mo><mi>y</mi></mrow></msub><mrow><mi>P</mi><mo stretchy=\"false\">(</mo><mi>c</mi><mi mathvariant=\"normal\">∣</mi><mi>x</mi><mo stretchy=\"false\">)</mo><mi>P</mi><mo stretchy=\"false\">(</mo><mi>c</mi><mi mathvariant=\"normal\">∣</mi><mi>z</mi><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">P(err)=1- \\sum _{c\\in y}{P(c|x)P(c|z)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">e</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">r</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.185818em;vertical-align:-0.43581800000000004em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.07765999999999995em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">c</span><span class=\"mrel mtight\">∈</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.43581800000000004em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">c</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\">x</span><span class=\"mclose\">)</span><span class=\"mord mathnormal\" style=\"margin-right:0.13889em;\">P</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\">c</span><span class=\"mord\">∣</span><span class=\"mord mathnormal\" style=\"margin-right:0.04398em;\">z</span><span class=\"mclose\">)</span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"<strong>最近邻分类虽然简单，但它的泛化错误率不超过贝叶斯最优分类器错误率的两倍！</strong>"}]},{"t":"list_item","d":5,"p":{},"v":"随着属性数量变多，现实任务中难以满足任意测试样本x附件的任意小的/delta距离范围内总能找到一个训练样本，即密采样","c":[{"t":"list_item","d":7,"p":{},"v":"维数灾难： 某一位空间的[0,1]区间里有5个训练样本，以相同的密度再d维空间里配置相同种类的训练样本的话，最终的样本数就达到了5…^d个。"},{"t":"list_item","d":7,"p":{},"v":"高维情况下出现的数据样本稀疏，距离计算困难"}]},{"t":"list_item","d":5,"p":{},"v":"低维嵌入","c":[{"t":"list_item","d":7,"p":{},"v":"将高维属性空间转变为一个低维“子空间”，子空间内，样本密度大幅提高，距离计算也变得更为容易"}]}]},{"t":"heading","d":3,"p":{},"v":"多维缩放：MDS"},{"t":"heading","d":3,"p":{},"v":"主成分分析：","c":[{"t":"list_item","d":5,"p":{},"v":"PCA","c":[{"t":"list_item","d":7,"p":{},"v":"对于正交属性空间中的样本点，用一个超平面对所有样本进行恰当的表达。"},{"t":"list_item","d":7,"p":{},"v":"超平面满足","c":[{"t":"list_item","d":9,"p":{},"v":"最近重构性"},{"t":"list_item","d":9,"p":{},"v":"最大可分性"},{"t":"list_item","d":9,"p":{},"v":"基于以上，可分别得到主成分分析的两种等价推导"}]}]},{"t":"list_item","d":5,"p":{},"v":"核化主成分分析（KPCA）"},{"t":"list_item","d":5,"p":{},"v":"鲁棒主成分分析（ RPCA）"}]},{"t":"heading","d":3,"p":{},"v":"流形学习"},{"t":"heading","d":3,"p":{},"v":"度量学习"}]},{"t":"heading","d":2,"p":{},"v":"作业","c":[{"t":"heading","d":3,"p":{},"v":"作业一","c":[{"t":"bullet_list","d":4,"p":{},"v":"","c":[{"t":"list_item","d":5,"p":{},"v":"题目1：","c":[{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi><mo>=</mo><mfrac><mrow><msubsup><mi>x</mi><mi>d</mi><mi>T</mi></msubsup><msub><mi>y</mi><mi>d</mi></msub></mrow><mrow><msubsup><mi>x</mi><mi>d</mi><mi>T</mi></msubsup><msub><mi>x</mi><mi>d</mi></msub></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">w=\\frac{x^T_d y_d}{x^T_d x_d}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.78965em;vertical-align:-0.636085em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.153565em;\"><span style=\"top:-2.606975em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8328928571428572em;\"><span style=\"top:-2.1527714285714286em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span><span style=\"top:-2.8448em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3472285714285714em;\"><span></span></span></span></span></span></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.3487714285714287em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15122857142857138em;\"><span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.5102em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.9190928571428572em;\"><span style=\"top:-2.214em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.286em;\"><span></span></span></span></span></span></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3448em;\"><span style=\"top:-2.3487714285714287em;margin-left:-0.03588em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15122857142857138em;\"><span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.636085em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"定义：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>x</mi><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><msub><mi>x</mi><mi>m</mi></msub><msup><mo stretchy=\"false\">)</mo><mi>T</mi></msup><mo separator=\"true\">,</mo><msub><mi>x</mi><mi>d</mi></msub><mo>=</mo><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mo separator=\"true\">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mo separator=\"true\">,</mo><mo>…</mo><mo separator=\"true\">,</mo><msub><mi>x</mi><mi>m</mi></msub><mo>−</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><msup><mo stretchy=\"false\">)</mo><mi>T</mi></msup></mrow><annotation encoding=\"application/x-tex\">x=(x_1,x_2,\\dots ,x_m)^T,x_d=(x_1-\\overline{x} ,x_2=\\overline{x} ,\\dots ,x_m-\\overline{x})^T</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0913309999999998em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.825em;vertical-align:-0.19444em;\"></span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.825em;vertical-align:-0.19444em;\"></span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"minner\">…</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.151392em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0913309999999998em;vertical-align:-0.25em;\"></span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8413309999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.13889em;\">T</span></span></span></span></span></span></span></span></span></span></span>，y同理"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi><mo>=</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><msub><mi>y</mi><mi>i</mi></msub><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">)</mo></mrow></mrow><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>x</mi><mi>i</mi></msub></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">w=\\frac{\\sum_{i=1}^{m}{y_i(x_i-\\overline{x})}}{\\sum^{m}_{i=1}{x_i^2}-\\overline{x}\\sum_{i=1}^{m}{x_i}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.6591869999999997em;vertical-align:-0.5991799999999999em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.060007em;\"><span style=\"top:-2.62642em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7046857142857144em;\"><span style=\"top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-2.8971428571428572em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32143857142857146em;\"><span></span></span></span></span></span></span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"></span><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8051142857142857em;\"><span style=\"top:-2.177714285714286em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-2.8448em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3222857142857143em;\"><span></span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span></span></span></span></span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"></span><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7046857142857144em;\"><span style=\"top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-2.8971428571428572em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32143857142857146em;\"><span></span></span></span></span></span></span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"></span><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.5350070000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32143857142857146em;\"><span></span></span></span></span></span></span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"></span><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span></span></span></span></span><span class=\"mclose mtight\">)</span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5991799999999999em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>y</mi><mi>i</mi></msub><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover></mrow></mrow><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><mo stretchy=\"false\">(</mo><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">)</mo></mrow></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">=\\frac{\\sum_{i=1}^{m}{(y_ix_i-y_i\\overline{x}}}{\\sum_{i=1}^{m}{(x_i^2=x_i\\overline{x})}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.6591869999999997em;vertical-align:-0.5991799999999999em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.060007em;\"><span style=\"top:-2.62642em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7046857142857144em;\"><span style=\"top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-2.8971428571428572em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32143857142857146em;\"><span></span></span></span></span></span></span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"></span><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8051142857142857em;\"><span style=\"top:-2.177714285714286em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-2.8448em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3222857142857143em;\"><span></span></span></span></span></span></span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span></span></span></span></span><span class=\"mclose mtight\">)</span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.5350070000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32143857142857146em;\"><span></span></span></span></span></span></span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"></span><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5991799999999999em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"又因为<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mover accent=\"true\"><mi>y</mi><mo stretchy=\"true\">‾</mo></mover><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mi>m</mi><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mover accent=\"true\"><mi>y</mi><mo stretchy=\"true\">‾</mo></mover><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mover accent=\"true\"><mi>y</mi><mo stretchy=\"true\">‾</mo></mover></mrow></mrow><annotation encoding=\"application/x-tex\">\\overline{y}\\sum_{i=1}^{m}{x_i}=\\overline{x}\\sum_{i=1}^{m}{y_i}=m\\overline{x}\\overline{y}=\\sum_{i=1}^{m}{\\overline{x}\\overline{y}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.104002em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.104002em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.825em;vertical-align:-0.19444em;\"></span><span class=\"mord mathnormal\">m</span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.104002em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mord overline\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"又有：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><msub><mi>x</mi><mi>i</mi></msub><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover></mrow><mo>=</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mo>∗</mo><mi>m</mi><mo>∗</mo><mfrac><mn>1</mn><mi>m</mi></mfrac><mo>∗</mo><msub><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow></msub><mi>m</mi><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mi>x</mi><msup><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">\\sum_{i=1}^{m}{x_i\\overline{x}}=\\overline{x}\\sum_{i=1}^{m}{x_i}=\\overline{x}*m*\\frac{1}{m}*\\sum_{i=1}{m}{x_i}=x\\overline{x}^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.104002em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.104002em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.63056em;vertical-align:0em;\"></span><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">∗</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.46528em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">m</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">∗</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.190108em;vertical-align:-0.345em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">∗</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0497100000000001em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.16195399999999993em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">m</span></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.834568em;vertical-align:0em;\"></span><span class=\"mord mathnormal\">x</span><span class=\"mord\"><span class=\"mord overline\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.63056em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">x</span></span></span><span style=\"top:-3.55056em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line\" style=\"border-bottom-width:0.04em;\"></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.834568em;\"><span style=\"top:-3.08346em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"得： $w=\\frac{\\sum_{i=1}^{m}{y_ix_i-y_i\\overline{x}-x_i\\overline{y}+\\overline{x}\\overline{y}}}{\\sum_{i=1}^{m}{x_i^2-x_i\\overline{x}-x_i\\overline{x}+\\overline{x}^2}$"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><mover accent=\"true\"><mi>y</mi><mo stretchy=\"true\">‾</mo></mover><mo stretchy=\"false\">)</mo></mrow><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mover accent=\"true\"><mi>x</mi><mo stretchy=\"true\">‾</mo></mover><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup></mrow></mrow></mfrac></mrow><annotation encoding=\"application/x-tex\">=\\frac{\\sum_{i=1}^{m}(x_i-\\overline{x})(y_i-\\overline{y})}{\\sum_{i=1}^{m}{(x_i-\\overline{x})^2}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.6300139999999999em;vertical-align:-0.5700069999999999em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.060007em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7046857142857144em;\"><span style=\"top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-2.8971428571428572em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32143857142857146em;\"><span></span></span></span></span></span></span><span class=\"mspace mtight\" style=\"margin-right:0.19516666666666668em;\"></span><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span></span></span></span></span><span class=\"mclose mtight\"><span class=\"mclose mtight\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7463142857142857em;\"><span style=\"top:-2.786em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.5350070000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mop mtight\"><span class=\"mop op-symbol small-op mtight\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.1785614285714283em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">m</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.32143857142857146em;\"><span></span></span></span></span></span></span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord overline mtight\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">x</span></span></span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span></span></span></span></span><span class=\"mclose mtight\">)</span><span class=\"mopen mtight\">(</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3280857142857143em;\"><span style=\"top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mbin mtight\">−</span><span class=\"mord overline mtight\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6755600000000002em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span></span></span><span style=\"top:-3.57756em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"overline-line mtight\" style=\"border-bottom-width:0.049em;\"></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.19444em;\"><span></span></span></span></span></span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.5700069999999999em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span></span></span></span>"}]}]},{"t":"bullet_list","d":4,"p":{},"v":"","c":[{"t":"list_item","d":5,"p":{},"v":"题目2：","c":[{"t":"list_item","d":7,"p":{},"v":"最小二乘法拟合：y = wx","c":[{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mo stretchy=\"false\">(</mo><mi>w</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></msubsup><mrow><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><mi>w</mi><msub><mi>x</mi><mi>i</mi></msub><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup></mrow></mrow><annotation encoding=\"application/x-tex\">L_1(w)=\\sum_{i=1}^{3}{(y_i-wx_i)^2}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">L</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.253718em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.954008em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>=</mo><mo stretchy=\"false\">(</mo><mn>0.9</mn><mo>−</mo><mi>w</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy=\"false\">(</mo><mn>0.1</mn><mo>−</mo><mn>0</mn><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy=\"false\">(</mo><mn>2.2</mn><mo>−</mo><mn>2</mn><mi>w</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">=(0.9-w)^2+(0.1-0)^2+(2.2-2w)^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mord\">.</span><span class=\"mord\">9</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mord\">.</span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\">0</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\">.</span><span class=\"mord\">2</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"通过将导数设置为0，得到："},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mfrac><mrow><mi mathvariant=\"normal\">/</mi><mi>p</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>l</mi><msub><mi>L</mi><mn>1</mn></msub><mo stretchy=\"false\">(</mo><mi>w</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi mathvariant=\"normal\">∂</mi><mi>w</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>2</mn><mo>∗</mo><mo stretchy=\"false\">(</mo><mn>0.9</mn><mo>−</mo><mn>2</mn><mo stretchy=\"false\">)</mo><mo>−</mo><mn>4</mn><mo>∗</mo><mo stretchy=\"false\">(</mo><mn>2.2</mn><mo>−</mo><mn>2</mn><mi>w</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">\\frac{/partial L_1(w)}{\\partial w}=-2*(0.9-2)-4*(2.2-2w)=0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.355em;vertical-align:-0.345em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.01em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\" style=\"margin-right:0.05556em;\">∂</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">/</span><span class=\"mord mathnormal mtight\">p</span><span class=\"mord mathnormal mtight\">a</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">r</span><span class=\"mord mathnormal mtight\">t</span><span class=\"mord mathnormal mtight\">i</span><span class=\"mord mathnormal mtight\">a</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">L</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31731428571428577em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mopen mtight\">(</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">2</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">∗</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mord\">.</span><span class=\"mord\">9</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">2</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">4</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">∗</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\">.</span><span class=\"mord\">2</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">0</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi><mo>=</mo><mn>1.06</mn></mrow><annotation encoding=\"application/x-tex\">w=1.06</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span><span class=\"mord\">.</span><span class=\"mord\">0</span><span class=\"mord\">6</span></span></span></span>"}]},{"t":"list_item","d":7,"p":{},"v":"使用：y=wx2","c":[{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mo stretchy=\"false\">(</mo><mi>w</mi><mo stretchy=\"false\">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></msubsup><mrow><mo stretchy=\"false\">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><mi>w</mi><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup></mrow><mo>=</mo><mo stretchy=\"false\">(</mo><mn>0.9</mn><mo>−</mo><mi>w</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy=\"false\">(</mo><mn>0.1</mn><mo>−</mo><mn>0</mn><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy=\"false\">(</mo><mn>2.2</mn><mo>−</mo><mn>4</mn><mi>w</mi><msup><mo stretchy=\"false\">)</mo><mn>2</mn></msup></mrow><annotation encoding=\"application/x-tex\">L_2(w)=\\sum^{3}_{i=1}{(y_i-wx_i^2)^2}=(0.9-w)^2+(0.1-0)^2+(2.2-4w)^2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\">L</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.253718em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.954008em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">i</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">y</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31166399999999994em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-2.441336em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.258664em;\"><span></span></span></span></span></span></span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mord\">.</span><span class=\"mord\">9</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mord\">.</span><span class=\"mord\">1</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\">0</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\">.</span><span class=\"mord\">2</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.064108em;vertical-align:-0.25em;\"></span><span class=\"mord\">4</span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\"><span class=\"mclose\">)</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141079999999999em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"对w求导为0："},{"t":"list_item","d":9,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mfrac><mrow><mi mathvariant=\"normal\">∂</mi><msub><mi>L</mi><mn>2</mn></msub><mo stretchy=\"false\">(</mo><mi>w</mi><mo stretchy=\"false\">)</mo></mrow><mrow><mi mathvariant=\"normal\">∂</mi><mi>w</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mn>2</mn><mo>∗</mo><mo stretchy=\"false\">(</mo><mn>0.9</mn><mo>−</mo><mi>w</mi><mo stretchy=\"false\">)</mo><mo>−</mo><mn>8</mn><mo>∗</mo><mo stretchy=\"false\">(</mo><mn>2.2</mn><mo>−</mo><mn>4</mn><mi>w</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">\\frac{\\partial L_2(w)}{\\partial w} = -2*(0.9-w)-8*(2.2-4w)=0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.355em;vertical-align:-0.345em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.01em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\" style=\"margin-right:0.05556em;\">∂</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.485em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\" style=\"margin-right:0.05556em;\">∂</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">L</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.31731428571428577em;\"><span style=\"top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.143em;\"><span></span></span></span></span></span></span><span class=\"mopen mtight\">(</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose mtight\">)</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">2</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">∗</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">0</span><span class=\"mord\">.</span><span class=\"mord\">9</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">8</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">∗</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\">2</span><span class=\"mord\">.</span><span class=\"mord\">2</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\">4</span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">0</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"得：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>w</mi><mo>=</mo><mfrac><mn>19.4</mn><mn>34</mn></mfrac><mo>≈</mo><mn>0.57</mn></mrow><annotation encoding=\"application/x-tex\">w=\\frac{19.4}{34}\\approx 0.57</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.190108em;vertical-align:-0.345em;\"></span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.845108em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">3</span><span class=\"mord mtight\">4</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1</span><span class=\"mord mtight\">9</span><span class=\"mord mtight\">.</span><span class=\"mord mtight\">4</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">≈</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">0</span><span class=\"mord\">.</span><span class=\"mord\">5</span><span class=\"mord\">7</span></span></span></span>"}]}]}]}]},{"t":"heading","d":3,"p":{},"v":"作业2","c":[{"t":"list_item","d":5,"p":{},"v":"题目1：","c":[{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>E</mi><mi>n</mi><mi>t</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>−</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>y</mi></msubsup><mrow><msub><mi>p</mi><mi>k</mi></msub><mi>l</mi><mi>o</mi><msub><mi>g</mi><mn>2</mn></msub><msub><mi>p</mi><mi>k</mi></msub></mrow></mrow><annotation encoding=\"application/x-tex\">Ent(D)=-\\sum^{y}_{k=1}{p_klog_2p_k}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">t</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.104002em;vertical-align:-0.29971000000000003em;\"></span><span class=\"mord\">−</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.804292em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029000000000005em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">y</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord mathnormal\" style=\"margin-right:0.01968em;\">l</span><span class=\"mord mathnormal\">o</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord mathnormal\">p</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.33610799999999996em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span></span>"},{"t":"list_item","d":7,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>G</mi><mi>a</mi><mi>i</mi><mi>n</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo separator=\"true\">,</mo><mi>a</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>E</mi><mi>n</mi><mi>t</mi><mo stretchy=\"false\">(</mo><mi>D</mi><mo stretchy=\"false\">)</mo><mo>−</mo><msubsup><mo>∑</mo><mrow><mi>v</mi><mo>=</mo><mn>1</mn></mrow><mi>V</mi></msubsup><mrow><mo stretchy=\"false\">(</mo><mfrac><msup><mi>D</mi><mi>v</mi></msup><mi>D</mi></mfrac><mi>E</mi><mi>n</mi><mi>t</mi><mo stretchy=\"false\">(</mo><msup><mi>D</mi><mi>v</mi></msup><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></mrow></mrow><annotation encoding=\"application/x-tex\">Gain(D,a)=Ent(D)-\\sum^{V}_{v=1}{(\\frac{D^v}{D}Ent(D^v))}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">G</span><span class=\"mord mathnormal\">a</span><span class=\"mord mathnormal\">i</span><span class=\"mord mathnormal\">n</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord mathnormal\">a</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">t</span><span class=\"mopen\">(</span><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.326231em;vertical-align:-0.345em;\"></span><span class=\"mop\"><span class=\"mop op-symbol small-op\" style=\"position:relative;top:-0.0000050000000000050004em;\">∑</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.981231em;\"><span style=\"top:-2.40029em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span><span class=\"mrel mtight\">=</span><span class=\"mord mtight\">1</span></span></span></span><span style=\"top:-3.2029em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.22222em;\">V</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.29971000000000003em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mopen nulldelimiter\"></span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.91098em;\"><span style=\"top:-2.6550000000000002em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span></span></span></span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"></span></span><span style=\"top:-3.394em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.7385428571428572em;\"><span style=\"top:-2.931em;margin-right:0.07142857142857144em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.345em;\"><span></span></span></span></span></span><span class=\"mclose nulldelimiter\"></span></span><span class=\"mord mathnormal\" style=\"margin-right:0.05764em;\">E</span><span class=\"mord mathnormal\">n</span><span class=\"mord mathnormal\">t</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02778em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.664392em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03588em;\">v</span></span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mclose\">)</span></span></span></span></span>"}]},{"t":"list_item","d":5,"p":{},"v":"题目2：","c":[{"t":"list_item","d":7,"p":{},"v":"（1）：","c":[{"t":"list_item","d":9,"p":{},"v":"+1","c":[{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub><mo>+</mo><mi>b</mi><mo>&lt;</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">w_1+w_2+b &lt;= 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73354em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&lt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span>"},{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>w</mi><mn>2</mn></msub><mo>&lt;</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">2w_1+2w_2 &lt;= 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&lt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span>"},{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><mi>b</mi><mo>&lt;</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">2w_1 + b&lt;= 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73354em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&lt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span>"}]},{"t":"list_item","d":9,"p":{},"v":"-1","c":[{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>b</mi><mo>&gt;</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">b &gt;= -1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73354em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">1</span></span></span></span>"},{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><mi>b</mi><mo>&gt;</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">w_1 + b &gt;= -1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73354em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">1</span></span></span></span>"},{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>w</mi><mn>2</mn></msub><mo>+</mo><mi>b</mi><mo>&gt;</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">w_2 + b &gt;= -1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.73354em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">1</span></span></span></span>"}]},{"t":"list_item","d":9,"p":{},"v":"得到：<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub><mo>&gt;</mo><mo>=</mo><mn>4</mn></mrow><annotation encoding=\"application/x-tex\">w_1 + w_2 &gt;= 4</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6891em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">4</span></span></span></span> 其中 <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>&gt;</mo><mo>=</mo><mn>2</mn><mo separator=\"true\">,</mo><msub><mi>w</mi><mn>2</mn></msub><mo>&gt;</mo><mo>=</mo><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">w_1 &gt;= 2, w_2 &gt;= 2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6891em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"></span><span class=\"mord\">2</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">2</span></span></span></span>求最小则<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><mo separator=\"true\">;</mo><msub><mi>w</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">w_1=2;w_2=2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.58056em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.8388800000000001em;vertical-align:-0.19444em;\"></span><span class=\"mord\">2</span><span class=\"mpunct\">;</span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.02691em;\">w</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">2</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"可得<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>b</mi><mo>&gt;</mo><mo>=</mo><mo>−</mo><mn>3</mn><mtext>且</mtext><mi>b</mi><mo>&lt;</mo><mo>=</mo><mo>−</mo><mn>3</mn><mtext>推出</mtext><mi>b</mi><mo>=</mo><mo>−</mo><mn>3</mn></mrow><annotation encoding=\"application/x-tex\">b &gt;= -3 且 b &lt;= -3 推出 b = -3</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73354em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&gt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.77777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">3</span><span class=\"mord cjk_fallback\">且</span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">&lt;</span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.36687em;vertical-align:0em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.77777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">3</span><span class=\"mord cjk_fallback\">推</span><span class=\"mord cjk_fallback\">出</span><span class=\"mord mathnormal\">b</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.72777em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">3</span></span></span></span>"},{"t":"list_item","d":9,"p":{},"v":"所以最后得到","c":[{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">2x_1+2x_2-3 = 0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">3</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">0</span></span></span></span>式1"},{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>1</mn><mtext>式</mtext><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">2x_1+2x_2-3 = -1式2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">3</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.76666em;vertical-align:-0.08333em;\"></span><span class=\"mord\">−</span><span class=\"mord\">1</span><span class=\"mord cjk_fallback\">式</span><span class=\"mord\">2</span></span></span></span>"},{"t":"list_item","d":11,"p":{},"v":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><mn>3</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding=\"application/x-tex\">2x_1+2x_2-3 = 1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">+</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.79444em;vertical-align:-0.15em;\"></span><span class=\"mord\">2</span><span class=\"mord\"><span class=\"mord mathnormal\">x</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.30110799999999993em;\"><span style=\"top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span><span class=\"mbin\">−</span><span class=\"mspace\" style=\"margin-right:0.2222222222222222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">3</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"></span><span class=\"mord\">1</span></span></span></span>式3"}]}]},{"t":"list_item","d":7,"p":{},"v":"(2)","c":[{"t":"list_item","d":9,"p":{},"v":"分别将上述各点代入上式得2，3求得等式左右相等得为支持向量"}]}]}]}]}]})</script>
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